Here's one attempt to explain the inverse cube law for centrifugal force. It's not as snappy as I'd like:
The 1/r³ law comes from combining two effects:
(1) the centrifugal force is proportional to v²/r where v is the velocity normal to the radial direction, and
(2) conserving angular momentum L = mvr requires v ∝ 1/r.
Putting them together: v² ∝ 1/r² divided by r gives 1/r³.
But why is the centrifugal force proportional to v²/r? Well, I can explain that, but now it's getting pretty long.
